A cohomology associated to a 1- form In this question all objects are real analytic.(manifolds, differential forms..)
Assume that $M$ is a compact  manifold and $\alpha \in \Omega^{1}(M)$ is a one form.
We define a map $\phi:\Omega^{*}(M)\to \Omega^{*+1}(M)$ with wedge product; $\phi(\beta)=\alpha \wedge \beta$. Then $\phi \circ \phi=0$. Then we have a complex of vector spaces. So we naturally  obtain a  cohomology.

Is each cohomology, a finite dimensional vector space?

 A: Suppose that $\alpha$ is nowhere zero. A differential form $\beta$ satisfies $\alpha\wedge \beta=0$ just when $\beta=\alpha \wedge \gamma$ for some $\gamma$ by Cartan's lemma. So the cohomology vanishes, finite dimensional. On the other hand, take $\alpha=0$. Then the kernel is everything, the image nothing, so the quotient is everything, infinite dimensional.
A: The cohomology is finite-dimensional for a generic $1$-form. In fact, it has a fairly simple description.
Indeed, as Ben McKay pointed out, it is sufficient to check this locally near a generic singularity, which has the form $\alpha= A_{i,j} x^i dx^j$. We might as well write this as $\sum_j y_j dx^j$, where $y_i = \sum_i A_{i,j}x^i$. Any failure of exactness among analytic functions will occur already in polynomials, since we can just take the leading terms of the power series. So we can write a complex of modules
$$\mathbb R[y_1,\dots,y_n] \to \bigoplus_{i} \mathbb R[y_1,\dots,y_n]dx^i \to \dots \to \mathbb R[y_1,\dots y_n] dx^1\dots dx^n$$
with the arrows coming from wedging with $\alpha$.
This complex is the tensor product of the complexes $\mathbb R[y_i] \to \mathbb R[y_i]$, with the arrow coming from multiplication by $y_i$, for $i$ from $1$ to $n$. This complex has cohomology $0$ except for degree $1$, where it is $\mathbb R$. So the tensor product has cohomology $0$ except for degree $n$, where it is $\mathbb R$. This $\mathbb R$ comes from the fact that all forms produced by wedging with $\alpha$ vanish at $0$, whereas a general $1$-form need not vanish.
So the total cohomology is $0$ except in dimension $n$, where it is $\mathbb R^d$, $d$ the number of singular points.
For a $1$-form with special singularities, the complex is equivalent to the Koszul complex with respect to the coefficients of the $dx^i$.
